ABSTRACT
B B Bι = + (6)
0.139 0.5 0.00020813 330 3.5
R nm a c d
δ= = = = =
stretch angleV V V= + (7)
( ){ }201 exp( ) 1stretch eV D r rβ = − − − − (8) 2 4
0 0 1 ( ) 1 ( ) 2angle sertic
V k kθ θ θ θ θ = − + − (9)
1.421.10 9.10
1.8.10 120
0.9.10 0.754
r mD Nm m
Nmk k rad radθ
= =
= = °
= =
V VF r M r
θ θ
∂ ∂ = =∂ ∂ (10)
1 0 0212 3 2( ) ( )cos ijk ij
EA V Vh EI rθ
∂ ∂ = = ∂ ∂ (11)
: : : :T L E H K M H E S K= + = + (12)
1 1 2 21 2, ,2 2 2 2 h h h hδ δ δ δ
α α γ γ γ− +− < < − < < < < (13)
, . 2
F h h
α αδγ δ δ δ ± ±
A h
αξ δ=
A h
αξ δ=
z γ δ= (17b)
31 0
h Z
β βς ∂ ∂
+ =∂ ∂ (18)
3 3 1 0µ µi in b n b atz zh
± ± ±+ = = (19)
i ib b h z
β βς ∂ ∂
+ =∂ ∂ (20)
3 3 1 0µ µi in b n b atz zh
± ± ±+ = = (21)
ij ijn ijn ijlm U Ub c c c
h zββ βς ∂ ∂
= + +∂ ∂ (22)
ij ijn ijn ijlm V Vb c c zc
h zββ βς ∂ ∂
= + + ∂ ∂ (23)
(24)
1 ( )1 16
C zb E B
v
γ γ
= = +
+ ∑ (27)
B cos sin C cos sin cos sinϕ ϕ ϕ ϕ ϕ ϕ= = + − (30)
1112 2211 2 2 1112 2211 2 2j j j j j j j jB B cos sin C C cos sinϕ ϕ ϕ ϕ= = = = − (31)
B B cos sin C C cos sin cos sinϕ ϕ ϕ ϕ ϕ ϕ= = = = − (32)
B B cos sin C C cos sin cos sinϕ ϕ ϕ ϕ ϕ ϕ= = = = − (33)
( )211 11 2231 16 3 EN piδ ε ε= + (34)
( )222 11 2231 16 3 EN piδ ε ε= + (35)
12 121 16 3 EN piδ ε= (36)
1 768 EM v k vk v l
piδ = + + + (37)
1 768 EM vk v k v l
piδ = + + + (38)
1 768 EM ( v )k v l
piδ= + +
(39)
12 64 ( ) ( )LJV r r
σ σ ε
= − (44)
, ,r EA EI GJk k k L L Lθ ∅
= = = (48)
( ) 12 64 12( ) 6( )LJ dV rF dr r r r
ε σ σ = = − + (49)
kd k
θ = (50)
= (51)
3 1 0 1 3 0
16 3 0 0 1
N EN l
N
ε piδ ε
ε
= (52)
3 1 0 1 16 3 1 3 0 0
0 0 1 0
N
E
ε
ε δ pi ε
− =
(53)
6 3 / ENE
l σ pi δ ε ε
δ = = = (54)
6 3 / ENE
l σ pi δ ε ε
δ = = = (55)
2 32 3 /N EG
l τ pi δ ε ε
δ = = = (56)
6 3SWCNT EE l
pi δ = (57)
6 25.488 10 ; 0.147 ; 0/ .142N nmE nm L nmδ−= × = = (58)
ESWCNT = 1.71 TPa (59a)
The corresponding shear modulus can be calculated using Eq. (58), and is GSWCNT = 0.32 TPa (59b)
/ /NT
P AE L L
σ
ε = =
∆ (60)
TLG J
= (61)
0 ( ) ( )2 2NT NT t tA R Rpi = + − − (62a)
0 ( ) ( )2 2 2NT NT t tJ R Rpi = + − − (62b)
( ) ( )2 20 , ,NT out NT inA R t R tpi = + − − (63a) ( ) ( )4 40 , ,2 NT out NT inJ R t R t
pi = + − − (63b)
0 ,r w vdwU U U U U U∅= + + + +∑ ∑ ∑ ∑ ∑ (64) where Ur is for a bond stretch interaction, Uθ for a bond angle bending, Uϕ for a dihedral angle torsion, U
2 20 1 1( ) ( ) , 2 2r r r
U k r r k r= − = ∆ (65)
2 20 1 1( ) ( ) , 2 2
U k kθ θ θθ θ θ= − = ∆ (66)
21 ( ) ,
2t w r U U U k∅= + = ∆∅ (67)
1 1 1 , 2 2 2
A N N L EAU dL L EA EA L
= = = ∆∫ (68) where ΔL is the axial stretching deformation. The strain energy of a uniform beam under pure bending moment M (Fig. 3.14b) is
1 2 1 2 , 2 2
M M EI EIU dL EI L L
α α= = =∫ (69) Where α denotes the rotational angle at the ends of the beam. The strain energy of a uniform beam under pure torsion T (Fig. 3.14c) is
1 1 1 , 2 2 2
T T T L GJU dL GJ GJ L
β= = = ∆∫ (70) where Δβ is the relative rotation between the ends of the beam.
